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1. Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

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   Braun, Benjamin; Bruegge, Kaitlin; Kahle, Matthew (December 2023, Discrete Mathematics & Theoretical Computer Science)

   Symmetric edge polytopes are lattice polytopes associated with finite simplegraphs that are of interest in both theory and applications. We investigate thefacet structure of symmetric edge polytopes for various models of randomgraphs. For an Erd\H{o}s-Renyi random graph, we identify a thresholdprobability at which with high probability the symmetric edge polytope sharesmany facet-supporting hyperplanes with that of a complete graph. We alsoinvestigate the relationship between the average local clustering, also knownas the Watts-Strogatz clustering coefficient, and the number of facets forgraphs with either a fixed number of edges or a fixed degree sequence. We usewell-known Markov Chain Monte Carlo sampling methods to generate empiricalevidence that for a fixed degree sequence, higher average local clustering in aconnected graph corresponds to higher facet numbers in the associated symmetricedge polytope. 

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2. Matching Polytopes, Gorensteinness, and the Integer Decomposition Property

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   Eisley, Benjamin; Matsushita, Koji; Vindas-Meléndez, Andrés_R (April 2025, Graphs and Combinatorics)

   Abstract The matching polytope of a graphGis the convex hull of the indicator vectors of the matchings onG. We characterize the graphs whose associated matching polytopes are Gorenstein, and then prove that all Gorenstein matching polytopes possess the integer decomposition property. As a special case study, we examine the matching polytopes of wheel graphs and show that they arenotGorenstein, butdopossess the integer decomposition property. 

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3. A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

   https://doi.org/10.1088/1751-8121/ad54a8  

   Cantarella, Jason; Schumacher, Henrik; Shonkwiler, Clayton (July 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract We present a faster direct sampling algorithm for random equilateral closed polygons in three-dimensional space. This method improves on the moment polytope sampling algorithm of Cantarellaet al(2016J. Phys. A: Math. Theor.49275202) and has (expected) time per sample quadratic in the number of edges in the polygon. We use our new sampling method and a new code for computing invariants based on the Alexander polynomial to investigate the probability of finding unknots among equilateral closed polygons. 

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4. Associahedra as moment polytopes

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   Gekhtman, Michael; Thomas, Hugh (December 2024, Bulletin of the London Mathematical Society)

   Abstract Generalized associahedra are a well‐studied family of polytopes associated with a finite‐type cluster algebra and a choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani‐Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster ‐variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction. 

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5. Towards instance-wise calibration: local amortized diagnostics and reshaping of conditional densities (LADaR)

   https://doi.org/10.1088/2632-2153/ae1f05  

   Dey, Biprateep; Zhao, David; Andrews, Brett_H; Newman, Jeffrey_A; Izbicki, Rafael; Lee, Ann_B (December 2025, Machine Learning: Science and Technology)

   Abstract Key science questions, such as galaxy distance estimation and weather forecasting, often require knowing the full predictive distribution of a target variableYgiven complex inputsX. Despite recent advances in machine learning and physics-based models, it remains challenging to assess whether an initial model is calibrated for allx, and when needed, to reshape the densities ofytoward ‘instance-wise’ calibration. This paper introduces the local amortized diagnostics and reshaping of conditional densities (LADaR) framework and proposes a new computationally efficient algorithm (Cal-PIT) that produces interpretable local diagnostics and provides a mechanism for adjusting conditional density estimates (CDEs).Cal-PITlearns a single interpretable local probability–probability map from calibration data that identifies where and how the initial model is miscalibrated across feature space, which can be used to morph CDEs such that they are well-calibrated. We illustrate the LADaR framework on synthetic examples, including probabilistic forecasting from image sequences, akin to predicting storm wind speed from satellite imagery. Our main science application involves estimating the probability density functions of galaxy distances given photometric data, whereCal-PITachieves better instance-wise calibration than all 11 other literature methods in a benchmark data challenge, demonstrating its utility for next-generation cosmological analyzes⁹9Code available as a Python package here:https://github.com/lee-group-cmu/Cal-PIT.. 

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